A finite field is a field containing only a finite number of elements, which is widely used in various engineering fields. At present, based on different design basis, multiplication over a finite field are mainly divided into four types: multiplication on the standard basis, multiplication on normal basis, multiplication on double basis, and multiplication on triangular basis.
A composite finite field is a special form of the finite field, and the composite finite field GF((2n)m) is the isomorphic form of the finite field GF(2n×m), which is effectively used in various cryptographic applications and encoding techniques. Effective multiplication design over the composite finite field plays a vital role in the implementation of cryptographic algorithms. There are a variety of known multipliers over composite finite fields in prior art, including software multiplier and hardware multiplier, both of which are devices for performing multiplication of two operands.
The multiplication of three operands is widely used in solving mathematical problems and engineering fields, for example, solving of the Oil and Vinegar polynomial which is commonly used in the cryptographic field. The structure of the Oil and Vinegar polynomial includes a plurality of multiplications of three operands as follows:
            ∑                        i          ∈                      O            l                          ,                  j          ∈                      S            l                                ⁢                  α        ij            ⁢              x        i            ⁢              x        j              +            ∑              i        ,                  j          ∈                      S            l                                ⁢                  β        ij            ⁢              x        i            ⁢              x        j              +            ∑              i        ∈                  S                      l            +            1                                ⁢                  γ        i            ⁢              x        i              +      η    .  
The Oil and Vinegar polynomial is the most common form of polynomial in multivariate public key cryptosystem. Each individual element of this polynomial is an element of the computing domain. When calculating the value of the Oil and Vinegar polynomial, especially the first two terms of αijxixj and βijxixj, the multiplication of three operands may be used for many times. The multiplication of three operands is not limited to this.
The existing techniques for solving the multiplication of three operands are realized by multipliers of two operands. However, under real-time and speed-sensitive circumstances, there is a need to use specific hardware devices to implement multiplication of three operands.